Random Walk Models for Stock Prices Statistics and Data Analysis Professor William Greene Stern School of Business Department of IOMS Department of Economics.

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Presentation on theme: "Random Walk Models for Stock Prices Statistics and Data Analysis Professor William Greene Stern School of Business Department of IOMS Department of Economics."— Presentation transcript:

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Random Walk Models for Stock Prices Statistics and Data Analysis Professor William Greene Stern School of Business Department of IOMS Department of Economics

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A Model for Stock Prices Preliminary: Consider a sequence of T random outcomes, independent from one to the next, Δ 1, Δ 2,…, Δ T. (Δ is a standard symbol for change which will be appropriate for what we are doing here. And, well use t instead of i to signify something to do with time.) Δ t comes from a normal distribution with mean μ and standard deviation σ. 1/30

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Random Walk Models for Stock Prices Application Suppose P is sales of a store. The accounting period starts with total sales = 0 On any given day, sales are random, normally distributed with mean μ and standard deviation σ. For example, mean $100,000 with standard deviation $10,000 Sales on any given day, day t, are denoted Δ t Δ 1 = sales on day 1, Δ 2 = sales on day 2, Total sales after T days will be Δ 1 + Δ 2 +…+ Δ T Therefore, each Δ t is the change in the total that occurs on day t. 2/30

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Random Walk Models for Stock Prices Random Walk Model Controversial – many assumptions Normality is inessential – we are summing, so after 25 periods or so, we can invoke the CLT. The assumption of period to period independence is at least debatable. The assumption of unchanging mean and variance is certainly debatable. The additive model allows negative prices. (Ouch!) The model when applied is usually based on logs and the lognormal model. To be continued … 13/30

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Random Walk Models for Stock Prices Observations - 1 The lognormal model (lognormal random walk) predicts that the price will always take the form P T = P 0 e ΣΔ t This will always be positive, so this overcomes the problem of the first model we looked at. 29/30

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Random Walk Models for Stock Prices Observations - 2 The lognormal model has a quirk of its own. Note that when we formed the prediction interval for P 25 based on P 0 = 40, the interval is [32.88,48.66] which has center at 40.77 > 40, even though μ = 0. It looks like free money. Why does this happen? A feature of the lognormal model is that E[P T ] = P 0 exp(μ T + ½σ T 2 ) which is greater than P 0 even if μ = 0. Philosophically, we can interpret this as the expected return to undertaking risk (compared to no risk – a risk premium). On the other hand, this is a model. It has virtues and flaws. This is one of the flaws. 30/30